Dark-field optical fault inspection of ∼10 nm scale room-temperature silicon single-electron transistors

Dark-field (DF) optical microscopy, combined with optical simulation based on modal diffraction theory for transverse electric polarized white light, is shown to provide non-invasive, sub-wavelength geometrical information for nanoscale etched device structures. Room temperature (RT) single electron transistors (SETs) in silicon, defined using etched ∼10 nm point-contacts (PCs) and in-plane side gates, are investigated to enable fabrication fault detection. Devices are inspected using scanning electron microscopy, bright-field (BF) and DF imaging. Compared to BF, DF imaging enhances contrast from edge diffraction by ×3.5. Sub-wavelength features in the RT SET structure lead to diffraction peaks in the DF intensity patterns, creating signatures for device geometry. These features are investigated using a DF line scan optical simulation approximation of the experimental results. Dark field imaging and simulation are applied to three types of structures, comprising successfully-fabricated, over-etched and interconnected PC/gate devices. Each structure can be identified via DF signatures, providing a non-invasive fault detection method to investigate etched nanodevice morphology.

RT operation of SETs requires quantum dots (QDs) < 10 nm in size to store charge, such that the QD capacitance C ∼ 1 aF or less and the single-electron charging energy E C = e 2 /2 C ? kT = 26 meV, the thermal energy at temperature T = 290 K [1,14].SET fabrication in the laboratory is often performed using electron beam lithography (EBL) [3] or scanning probe lithography (SPL) [15].The small size of RT SETs results in nanofabrication challenges in lithography, pattern transfer and structural uniformity.In additional fabrication steps, e.g.over-etching during pattern transfer, metal lift-off failure, resist development failure, etc, problems may also occur.These fabrication problems are more significant for semiconductor device fabrication at the ∼10 nm scale Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.[16][17][18].Hence, device structural metrology is essential for fabrication fault detection and process optimisation.
Typically, inspection of semiconductor nanodevices at various stages of the fabrication process are conducted using scanning electron microscope (SEM) or atomic force microscope (AFM) imaging [3,19].Scanning electron microscopy offers few or even sub-nanometre resolution [20,21], and edge and material contrast [22,23].However, SEM inspection may be invasive, causing charging [24][25][26] and carbon contamination [27][28][29] problems on inspected samples which lead to threshold voltage shifts in field-effect transistor (FET) type devices [30,31].These problems are particularly significant in SETs, where the single-electron current-voltage (I-V ) characteristics can be switched by even fractional changes in charge e nearby the QD [1].Atomic force microscopy, while also having few-nanometre resolution, additionally provides information on the vertical geometry of the sample.However, imaging is relatively slow and can also be invasive, e.g.tip induced damage and contamination can alter the device characteristics [32,33].
Non-invasive metrology, e.g.optical methods, may allow fabrication fault detection without altering nanodevice electrical characteristics.Previous work on nanoelectromechanical systems (NEMS) structures has shown that sub-wavelength features can be imaged by non-invasive optical microscopic imaging.The observed line scans can be predicted by applying a rigorous modal diffraction theory [34][35][36][37][38] to transverse electric (TE) white light, extending the work by Hopkins [39], Nyssonen [40,41], Sheridan [42][43][44] and Syms [38,45].Of particular interest, etched NEMS structures [45] have shown high contrast in dark-field (DF) optical imaging.It has also been demonstrated that DF microscopy allows imaging of ∼10 nm width gold nanorods, due to enhancement of the electric field by localised surface plasmon resonance nanoparticles [46].These methods provide signatures of the underlying sub-wavelength nanostructures in the DF image.In the DF fault detection method reported here, the mismatches between the signatures for successfully fabricated result, generated by modal diffraction theory, and the imaging result suggest errors in fabrication results.Thus, a known recipe is required for this method.The complexity of the interference patterns inhibits direct transfer of fault signatures between different structures.However, the combination of DF imaging and optical modal diffraction theory simulation is applicable for fault detection in other structures including asymmetric structures, provided that the structure can be represented in an approximate manner using Fourier Series.
In this work we have applied DF imaging methods to deep sub-wavelength (∼10 nm) silicon PC RT SET etched structures, to develop a method for fabrication fault detection.These devices consist of a SiO 2 PC region incorporating dopant atom QDs, connecting to triangular n + silicon source/ drain regions.Two trench-isolated, in-plane side gates are defined on either side of the PC.The devices have been inspected using SEM, bright-field (BF) and DF imaging.Compared to BF imaging, DF imaging has been shown to enhance contrast by ×3 due to the edge diffraction of collected light.Furthermore, sub-wavelength, nanoscale features in the RT SET structure lead to diffraction peaks in the DF image intensity, defining signatures of device geometry.These features may be explained using optical simulation of DF line scans arising from 2D vertical cross-section approximations of structures corresponding to (a) successfully fabricated, (b) over-etched (nanogap) and (c) failed lift-off (interconnected gates and PC) devices.It is shown that each structure can be identified using the DF microscopy and its signature using optical simulation, hence providing a method to detect fabrication faults in etched nanodevices.

Device fabrication and characterisation
A scanning electron micrograph of a Si-SiO 2 -Si PC RT SET, representative of the devices investigated in this study, is shown in figure 1(a).The device is defined in the heavilydoped (n-type, P dopant, concentration ∼10 20 cm −3 ) 40 nm thick top silicon (Si) layer of silicon-on-insulator (SOI) material (1 μm thick buried oxide (BOx) layer on a Si substrate).The fabrication process (see supplementary information) uses a high-resolution EBL (Raith-Vistec VB6-UHR machine), to expose a bilayer poly-methyl methacrylate (PMMA) resist which defines an Al (35 nm thick) hard-mask (figures 1(b)-(i)) by lift-off.Reactive ion etching (RIE) in SF 6 :O 2 (3:1) is used to transfer the pattern into the top Si layer and partially into the underlying SiO 2 to define dopant atom QDs suitable for RT operation (figure 1(c)).Figure 1(b) shows schematically the device into the BOx layer, for trench isolation of the PC (figures 1(b)-(ii), (iii)).Geometric oxidation [13] is used to fully oxidise the PC region, isolating phosphorus (P) dopant atoms in the oxide, with the complete device structure shown in figures 1(b)-(iv).Single-electron charging of the QDs and the drain-PC-source current paths can be tuned by the side-gate voltages.
The drain (D), source (S) and the two in-plane side-gates (G1 and G2) are indicated on the scanning electron micrograph in figure 1(a).The point-contact region, of width and length ∼10 nm and height ∼200 nm, lies between the drain and source and is separated from the gates on each side by ∼100 nm.Narrow projections at the tip of the gates increase the electrostatic coupling to the point-contact, while maximising the area in-between the gates and point-contact for better lift-off of an aluminium hard mask for pattern transfer into the Si (figures 1(b)-(ii).The rougher, etched areas are the exposed BOx layer of the SOI material.
Successful operation of a device requires the formation of point-contact between the source and drain regions, trench isolation of the double gates from the point-contact region, point-contact dimension ∼10 nm × 10 nm and an etched trench depth deeper than 40 nm.Lithographical problems, e.g.failed hard mask lift-off, can lead to interconnectivity between the gates and the point-contact region.Undercut during plasma etching can lead to removal of the PC region.Finally, an etch depth, which does not reach the BOx layer, leads to gate leakage current [47].Previous fabrication results have shown a fabrication failure rate of ∼11% with ∼37% showing SET electrical behaviour [47].
Scanning electron microscope (SEM) inspection can modify the device surface structure.Traces of a previous scan may be observed in figure 1(a), where a darker rectangular perimeter can be observed around the point-contact area.This perimeter defines the edges of the scan and may be caused by carbon deposition or charging, leading to changes in the SET electrical characteristics.Furthermore, SEM inspection implants charge into the BOx layer, acting as an 'offset' charge [48,49], which induces threshold voltage shifts and further changes the SET characteristics.Changes in the charge environment around the QD island of a SET can lead to shifts in the QD energy levels, a reduction in the Coulomb blockade effect, and introduce new current paths between the drain and the source.
The oxidised point-contact region is shown schematically in figure 1(c).As the source and drain regions on either side are wider, a Fermi sea of electrons can exist in the highly doped silicon.In contrast, impurity atoms in the point-contact area are isolated by tunnel barriers within the SiO 2 .For electrons to move from drain to the source, a current path tunnelling through a series of impurity atoms must form, (indicated by the red line).The impurity atoms behave as QDs, with single-electron charging and quantum confinement of energy states occurring within the potential well of the impurity atom [3].
The RT (RT = 290 K) drain-source current (I ds ) versus drain (V ds ) and gate voltage (V gs ) electrical characteristics of a point-contact SET are shown in figure 1(d).'Coulomb diamond' low current regions (marked by white solid lines) are observed around V ds = 0 V, indicating single electron charging of a dominant impurity atom QD [3].The Coulomb diamonds do not pinch-off to zero V ds width as V gs is varied, implying that a threshold value of |V ds | is necessary for current flow.This further suggests that a series potential barrier along the current path is present, attributable to additional impurity atoms/tunnel barriers with poor gate coupling.Figure 1(e) shows the SET equivalent circuit for this situation, where four tunnel barriers and three interleaved QDs are shown.eV where eV ds is the maximum width of the Coulomb diamond, minus the threshold voltage in V ds , and 13 aF is the total capacitance of QD2.We further assume C 2 = C 3 = 0.2 aF as a clear Coulomb staircase is not observed, implying similar tunnel barriers T 2 and T 3 .A similar argument implies that T 1 and T 4 are also similar, i.e.C 1 = C 4 .We use C 1 = C 4 = 0.05 aF to obtain the observed threshold voltage in V ds .Finally, while a symmetrical circuit exists around the central QD and only three QDs are used, further potential barriers and QDs may also exist along the drain source current path.

Optical imaging and simulation
A DF optical micrograph of a region containing 5 RT SETs is shown in figure 2(a) and the corresponding bright field (BF) image in figure 2(b).In figure 2(a), the dark areas are flat surfaces on the SOI chip, and the bright lines correspond to the edges of structures.In DF imaging, light is incident from an angle greater than the collection angle of the objective lens [50].Since the reflection angle is identical to the incident angle for flat surfaces, the light reflected from flat surfaces is not within the acceptance range of the objective lens and is therefore rejected.In contrast, when light is incident on the edges, edge diffraction produces waves travelling within the collection angle of the objective lens.Peaks in the image intensity are attributed to diffraction orders lying within the collection angle.For the experimental set-up of figure 2(a), the objective lens has a maximum collection angle of ±13°to vertical.Filtering out information from flat surfaces while collecting information from edges enhances edge contrast and facilitates the investigation of sub-wavelength SET features.Furthermore, due to the high concentration of edges at the SET point-contact areas, the greatest intensity of light is collected from these areas.In contrast, in the BF image (figure 2(b)), the intensity at the flat surface areas is significantly higher than the edges, which appear as dark lines.This is a consequence of the larger collection of reflected light from flat surfaces by the objective, preventing observation of edge diffraction effects.Furthermore, our areas of interest, i.e. the point-contact regions, include a greater concentration of edges and are much darker, hence becoming difficult to investigate.
A normalised greyscale line scan along the blue line in the DF image of figure 2  To obtain an understanding of the images 'rigorous modal diffraction theory' (RMDT) [34][35][36][37][38], based on wave propagation and diffraction mode analysis, was applied to white light imaging.Using RMDT simulations, the DF and BF patterns of PC SET structures, represented by 2D vertical slices, could be approximated.
Optical simulations using RMDT, where diffraction modes of a propagating plane wave are analysed, was applied to vertical slices of a white-light image obtained from the PC SET structure.These simulations were used to investigate the optical DF and BF patterns experimentally observed for the PC SET structure.This optical simulation proceeds by first solving for the propagation modes of the TE polarized waves assuming solution to Helmholtz equations for all layers, followed by matching boundary conditions and reconstruction of x-z polarised TE field.The optical simulation based on rigorous modal diffraction theory for TE polarized waves [38,45] solves for the x-z TE field diffraction modes to patterns for illumination and reflection (see supplementary material).Incident transmitted and reflected waves [38] are defined, with wave numbers k xLω (vertical direction), k zLω (horizontal direction, along substrate surface), the subscripts L and ω correspond to diffraction mode number and frequency.To solve for the diffraction modes L, the simulated structure is assumed to be periodic, with period Λ.In figure 3 The fundamental horizontal mode where q is the angle of the light to the vertical.Consideration of a repetitive 'grating' structure gives horizontal modes to higher orders, where L denotes the Lth diffraction order, and L is an integer.
Helmholtz equation for time-independent vector wave equation for TE field E is given by e 2 for all layers.After expanding this in diffraction orders L and matching same-order terms, the vertical mode numbers for layer 1 and layer 3 are given by e 2 where ε rn is the relative permittivity profile in the nth layer [38].ε for layer 1 and 3 is constant, due to consistency of medium throughout the layer.However, for layer 2, 2 cannot be solved directly by matching terms in the single electric field equation.Boundary conditions, imposed by layers 1 and 3, allow extraction of the solution through Floquet's theorem.Each mode consists of horizontal and vertical mode numbers of the electric field, their respective electric field amplitude scaling factors, and a Fourier series representation of ε r2 (n), where n is the order of Fourier series.A matrix M, which consists of the permutated Fourier series terms of ε r2 (n), the horizontal mode number k zL , and the fundamental wavenumber k 0 , can be solved for scaling factors in the z direction, and mode numbers in the xdirection.Compared to previous work on side-wall structures of single periodicity [38,45], our simulation is conducted by including multiple features with different widths (i.e.pointcontact and gate region widths) within the device layer.As four edges exist within the central point-contact region in (a), the patterns in (b), (c) are highly edge dependent.Finally, a more complex material variation is also considered as layers 2-3 contain Si, SiO 2 and air, in contrast to previous work conducted in simpler device/substrate layers (Si/air only) [45].
After solving for all the k xL , boundary conditions are applied to solve for scaling factors in the x-direction for all modes, across all layers.For x-z time-independent polarised electric field, the amplitude of the electric field and the partial differential along x, proportional to the magnetic field, match at the layer 1-2 and layer 2-3 boundaries.Boundary matching at each frequency and incident angle gives the equation Y = NX, where matrix N corresponds to the phase shift contribution from all x modes, and vector X consists of the scaling factors of modes across all layers.This allows calculation of scaling factor vector = - X N Y. 1 The final electric field amplitude is calculated using X and the modes k xL and k zL .At the end of the calculation for each incident angle at each wavelength, the electric field at each point is multiplied by its own conjugate, i.e.
and summed incoherently across all incident angles and wavelengths.Since where I is the intensity of light, the plotted simulated normalised power can be compared to the intensity of the DF images.The 2D simulation results of figures 4(b)-(c) use all x-z modes to calculate the resultant electric field energy.For later 1D DF simulation (figures 4-6), only diffraction modes with reflection angle (q = -( ) / k k sin L Z L 1 0 ) at Layer 1 within the objective lens collection angle (30°) are used to calculate the electric field energy.Hence, 2D simulation results give an overview of the BF/DF imaging in the x-z plane, and 1D DF simulation is considered using multiple wavelengths λ ∼ 0.4-0.7 μm across a range of incident angles, i.e. 35°-45°, where electric field power of all wavelengths and incident angles is summed incoherently for the resultant 1D electric field power, allowing creation of a DF signature.
Figures 3(b) and (c) show 2D BF and DF optical simulation results for |E y |, at λ = 0.4 μm, for the RT SET point-contact core.Simulations are conducted for a pointcontact width of 20 nm and 230 nm point-contact to gate separation.Diffraction fringes in |E y | are observed, with a period comparable to the simulation periodicity Λ.In the BF simulation (figure 3(b)), light is collected with a high amplitude from the flat surfaces compared to the point-contact area.In the DF simulation (figure 3(c)), light is incident between 35°and 45°from the left-hand side.Reflection from flat surfaces at around 40°can be observed, however this is rejected by the objective lens as it lies outside the collection angle (30°) of the objective lens.In contrast, light diffracted from the point-contact area lies within the collection angle and contributes to DF images.Furthermore, diffraction patterns formed around the point-contact areas have a different signature (narrower fringes) compared to the light reflected by flat surfaces.This implies that when combined with simulation, an experimental DF pattern can be analysed to determine the geometry of sub-wavelength features around the point-contact area.

Dark field optical device inspection of RT-SETs
Three specific cases of the application of DF inspection of RT-SETs are presented to highlight the application of the technique, with the resulting results analysed.These cases are (a) a SET that has been successfully fabricated, (b) a SET that has been over-etched, and (c) a SET where the lift-off process has failed. .This device provides a base-line DF image for a successfully fabricated device.Separate analysis of different colour channels in the image sensor can increase resolution, since colour channels detecting shorter wavelengths, i.e. blue channel with ∼400-500 nm, can detect more prominent double peaks in theory, hence an increase the resolution of detection.However, this can be very misleading for a range of reasons.There can be limitations in the microscope optics, e.g. the illumination source is neither single wavelength nor uniform across all wavelengths, there is a finite depth-of-field at large magnification, and finally the detector sensitivity is certainly not wavelength independent.All these factors can compromise the technique yet would be challenging to eliminate.Moreover, greyscale intensity analysis compensates the influences of non-uniform sensitivity of photo detectors across all wavelengths.
Comparison of SEM and DF images shows the resolution limit of optical microscopy.However, features generated by the underlying point-contact region can still be observed in the DF image.In this, a bright ring is formed at the pointcontact area in the DF image, (blue arrow).In addition,

SET with over-etched point-contact area
A SEM image of an over-etched RT-SET is shown in figure 5(a), where the point-contact has been lost due an excessive undercut during the RIE stage of the fabrication process, leaving a nanogap between source and drain regions.The effect of over-etching is also observed at the projecting tips of the gate electrodes.Figure 5(b) shows a DF optical image of the same device, providing a base-line DF image for over-etched devices.In this, the DF diffracted edge lines merge when approaching the point-contact area (blue arrow).Figure 5(c) shows a 2D DF modal diffraction simulation of the normalised electric field amplitude across a gate-nanogap-gate structure, corresponding to an over-etched RT-SET.The distance between the gates is 600 nm.A double peak (indicated by red arrows) is observed at the origin, i.e. the centre of the nanogap.For DF illumination of the SET centre area, due to the absence of the point-contact, fewer edges are available to diffract light to the objective lens.Therefore, the nanogap is barely resolved, leading to the shallow valley seen in figure 5(c).On the contrary, the point-contact in successfully fabricated devices provides extra edges, leading to greater diffraction and hence the more prominent double peak seen in figure 4  .This method was applied to extract the peak-valley differences for line scans in figure 5(f).The peak-valley difference of the line scans, normalised to the lower peak height, is plotted in figure 5(f).In this figure, a second-order curve fit shows that the peak-valley difference approaches zero when the line scans are more centralised to the nanogap.This trend, combined with DF modal diffraction theory simulation, provides a signature for over-etch devices.The variations of peakvalley difference of line scans around the point-contact region is unique among the three types of RT-SET fabrication results discussed, therefore can be used as a signature to identify over-etch SETs.

SET with failed lift-off in point-contact area
Figure 6(a) shows a SEM image of a device where lift-off has failed, leading to interconnections between the side gates and source-drain regions.The connection of electrodes at the point-contact area, attributed to the failed lift-off in the hardmask defining process, forms a central ∼600 nm square region in the top Si device layer.However, the connection of the top gate to the nominal point-contact region is weak, creating two notches ∼200 nm deep on each side of the projecting tip of the top gate, indicated by yellow arrows.In DF imaging, the presence of these notches introduces diffraction peaks.A DF optical image of the same device is shown in figure 6(b), where the interconnection of the gates/ source/drain regions is resolved (blue arrow).A weak line is present at the top gate connection due to the ∼200 nm notches in the top gate (figure 6(a)).We note that the widths of the central features in the DF image (b) are larger than in the SEM image because DF imaging provides a signature of the underlying geometric structure rather than the actual geometry for sub-wavelength imaging.Figure 6(c) shows a 2D DF modal diffraction theory simulation of the normalised electric field amplitude across the structure in figure 6(a).The simulated structure has an air gap of 200 nm that is located −600 nm relative to the origin, i.e. the centre of the pointcontact, which represents the air gaps on either side of G1 (orange arrows in figure 6(a)).This produces a single peak (indicated by a red arrow) centred at −600 nm in figure 6(c).Figure 6(d) shows normalised light intensity line scans, extracted from figure 6(b) between the white arrows.Large peaks are observed in lines 1-6 on the left-hand-side of the point-contact, centred at ∼−600 nm to the left of the centre of the point-contact.These peaks are ∼ 0.35 higher compared to a smaller right-hand-side peak, caused by higher-order diffraction fringes of the edge of G2 in figure 6(a).The boundary of the bottom gate (G2) is separated by ∼600 nm, where the main lobes of the DF diffraction are resolved but still connected.This connection of DF diffraction, from the boundary of G2, contributed to the right-hand-side peak in figure 6(d).This trend, combined with DF modal diffraction theory simulation, provide a signature for failed lift-off devices.

Conclusion
DF optical microscopy, combined with optical simulation based on rigorous modal diffraction theory, has been developed to provide non-invasive, sub-wavelength geometrical information about nanoscale etched device structures.This has been applied to silicon SETs, which incorporate etched ∼10 nm PCs and in-plane side gates.Devices have been inspected using SEM, BF and DF imaging and it has been shown that in comparison with BF, DF imaging enhances contrast from edge diffraction by ×3.Edge diffraction peaks observed in the DF intensity pattern of the sub-wavelength features in our SET structures, have established signatures which define the device geometry.These features have been explained by DF line scan optical simulation approximation.DF imaging has been applied to three types of structure, (a) those successfully fabricated, (b) over-etched and (c) interconnected PC/gate devices.Each has been identified by its DF signature, thus providing a non-invasive fault detection method for etched nanodevices.This combination of DF imaging and optical modal diffraction theory simulation can be applied to fault detection in other structures, including asymmetric structures, that is provided the structure can be represented realistically using a Fourier Series.

Figure 1 (
f) shows RT single-electron Monte Carlo simulations for the Coulomb diamond characteristics of this circuit, which qualitatively reflect the experimental behaviour.In figure 1(e), series barriers T 1 , T 4 , and QDs, QD 1 and QD 3 , uncoupled from V gs , are used on either side of the core SET, formed by T 2 , T 3 and QD 2 .The voltage drops across T 1 , T 4 , and charging of QD 1 and QD 2 leads to the experimentally observed threshold in V ds (figure1(d)) in the simulations.Coulomb diamonds are formed in the simulations by the core SET circuit.The gate capacitance C g = 0.73 aF used in the simulations may be extracted from the horizontal width of the diamonds,

Figure 1 .
Figure 1.(a) Scanning electron micrograph of a point-contact (PC) SET.The PC lies between the drain (D) and source (S) electrodes.Gate electrodes G1 and G2 are defined on either side of the PC.(b) Schematic diagram of the fabrication process for the PC RT-SET.(i) PMMA soft mask, is used to define an Al hard mask, (ii) pattern transfer using reactive ion etching (RIE), (iii) the transferred pattern in the top Si layer, and (iv) formation of tunnel barriers and dopant atom QD isolation using dry oxidation.(c) Schematic of the SET PC region.Grey area is SiO 2 and blue area is highly doped Si.Black dots indicate P dopant atoms, with circles in Si regions indicating Bohr radii.Dopant atoms in SiO 2 form QDs. (d) |I DS | versus V DS , V GS electrical characteristics of a RT-SET measured at 290 K. Black area shows Coulomb Blockade regions.Coulomb Diamonds are indicated by white lines.(e) SET equivalent circuit diagram for simulation in (f).(f) Simulation of |I DS | versus V DS and V GS characteristics of (e) using a single-electron Monte Carlo simulation.N.B., the close similarity between the experimental result and the simulation.
(a) is shown in figure 2(c).Similarly

Figure 2 .
Figure 2. Optical microscopy was undertaken using a Leica 'DMR Trinocular Fluorescence' microscope with a Leica 'PL FLUOTAR' 20×/ 0.45 objective lens.(a) DF optical image of a region containing five SETs.Edges are highlighted as bright lines and the point-contact areas can be seen as bright spots.(b) Same region, seen in bright field (BF).Edges appear as dark lines, but the point-contact area is dark and details within it cannot be resolved.Line scans (shown as blue lines) across images in (a) and (b) are presented in (c) and (d) respectively.(c) and (d) Normalised, greyscale line scans along the blue lines, plotted with pixel number on the x-axis.The peaks in DF (c) and valleys in BF (d) images correspond to edges along the SET features.Edge contrast, i.e. valley to peak difference, in the BF image is <0.21.In comparison, DF edge contrast, i.e. peak to valley difference is >0.75.

Figure 3 shows
(a) the layer structure, electric field magnitude with (b) BF illumination, and (c) DF illumination.In figure 3(a), Layer 1 is 5.0 μm thick and represents air (white) above the point-contact region.Layer 2 is 0.2 μm thick and corresponds to the top Si device layer.Grey represents the Si device topography around the point-contact region, and white represents the air gap between the gates and the point-contact.Layer 3 is 4.8 μm thick and corresponds to the BOx layer with SiO 2 in dark grey.A schematic representation of the objective lens is shown above the device layer.The simulated maximum collection angle of the objective lens is 30°, (Leica HC PL APO 100×/0.90BD objective lens), and is used for the DF analysis in figures 4-6.Since the light in conventional BF microscopy is illuminated vertically from the objective lens, only vertically incident light is considered in the BF modal diffraction theory simulation (indicated by the blue arrow in (a)).In contrast, in DF microscopy, light is incident at an angle greater than the maximum collection angle of the objective lens.Therefore, for DF modal diffraction theory simulation, incident light is simulated with angles in the range 35°-45°, in accordance with the experimental objective lens parameters.

Figure 3 .
Figure 3. (a) Device structure for optical modal diffraction theory simulations.Layer 1 is air (white), and layer 2 is the device layer, with Si (light grey) and air (white) regions.Layer 3 is the SiO 2 (dark grey) substrate.Λ indicates the grating period used in simulations, and a red dot marks the origin of the axis.(b) and (c) show the 2D modulus (colour scale) of the simulated x-z TE field for BF (b) and DF (c) illumination.The x and y axis values correspond to displacements relative to the point-contact centre.The air-point-contact centre interface lies at (0, 0) of the coordinate system.
(a), repetitive structures to obtain periodicity are shown on either side of the central pointcontact/side gate region.In the simulation, the etched device structure lies along the z-axis, while device layers are along the vertical x-axis.Patterns are generated for multiple wavelengths λ ∼ 0.4-0.7 μm and hence the fundamental wavenumber =

Figure 4 .
Figure 4. (a) Scanning electron micrograph of a RT-SET successfully fabricated.(b) High resolution SEM image.The point-contact is indicated by the yellow arrow.(c) DF image of the RT-SET shown in (a) and (b).The blue arrow shows the location of the point-contact region.A bright ring can be seen in the point-contact area.Line scans are extracted (shown in (e)) between the white arrows.The two yellow arrows indicate the location of the higher order diffraction fringes lying parallel to the edge of the drain (D) and source (S) electrodes.The SEM micrograph from (b), magnified to the same scale as this DF image, is represented on the top right hand side of the DF image.(d) DF optical theory simulation of a successfully fabricated device.A double peak (red arrows) with a deep valley between is observed.Yellow arrows indicate diffraction fringes in the simulation.(e) Line scans (1-6) across gate-point contact-gate were extracted vertically from grey scale intensity between the white arrows in the DF image in (c).Each line is displaced by one pixel in the DF image and normalised power is offset from the preceding line by 0.5.(f) Ratio of valley-peak difference to lower peak magnitude for line scans in (e), plotted with respect to line scan number.This ratio lies in the range of ∼0.2-0.3 for all scans.

Figure 5 .
Figure 5. (a) Scanning electron micrograph of an over-etched RT-SET, where the point-contact is missing.(b) DF image of the device in (a).The blue arrow shows the location of the missing point-contact region.Line scans are extracted (shown in (d)) between the white arrows.(c) DF modal diffraction theory simulation of a device with a missing point-contact.A double-peak (red arrows) with a shallow valley is observed.(d) Line scans (1-6) extracted from DF image in (b), across gate-nanogap-gate.Each line is displaced by one pixel in the DF image and normalised power is offset from the preceding line by 0.5.(e) The normalised power (top) and the first derivative of the normalised power (bottom).The red crosses in the normalised power plot indicate the location of the peak and the valley.The corresponding first derivative values of the valley and the right peak, indicated by red crosses in the bottom figure, show approximately zero values.(f) Ratio of peak-valley difference to right peak magnitude for line scans in (d), plotted with respect to line scan number.This ratio varies in the range ∼0-0.1.

4. 1 .
figure4(b).This device provides a base-line DF image for a successfully fabricated device.Separate analysis of different colour channels in the image sensor can increase resolution, since colour channels detecting shorter wavelengths, i.e. blue channel with ∼400-500 nm, can detect more prominent double peaks in theory, hence an increase the resolution of detection.However, this can be very misleading for a range of reasons.There can be limitations in the microscope optics, e.g. the illumination source is neither single wavelength nor uniform across all wavelengths, there is a finite depth-of-field at large magnification, and finally the detector sensitivity is certainly not wavelength independent.All these factors can compromise the technique yet would be challenging to eliminate.Moreover, greyscale intensity analysis compensates the influences of non-uniform sensitivity of photo detectors across all wavelengths.Comparison of SEM and DF images shows the resolution limit of optical microscopy.However, features generated by the underlying point-contact region can still be observed in the DF image.In this, a bright ring is formed at the pointcontact area in the DF image, (blue arrow).In addition,

Figure 6 .
Figure 6.(a) Scanning electron micrograph of a RT-SET structure having failed lift-off, leaving all the electrodes inter-connected.Yellow arrows indicate the boundary of a connected gate electrode.(b) DF image of the device in (a).The blue arrow shows the location of the failed lift-off region.Line scans are extracted (shown in (d)) between the white arrows.(c) DF modal diffraction theory simulation of a device with all connected electrodes.A single-major-peak (red arrow) is observed.(d) Line scans (1-6) extracted from DF image in (b), across gatenanogap-gate.Each line is displaced by two pixels in the DF image and normalised power is offset from the preceding line by 0.5.
Figure 4(e) shows normalised greyscale light intensity line scans extracted vertically from the measured DF image, for the region between the white arrows in figure 4(c), for comparison with simulation (d).The Gaussian background associated with the substrate and any stage tilt has been removed.Observation of a double peak in figure 4(e) is consistent with the simulation results in (d) for all line scans, although the central valley is less prominent than in the simulation.The peak-valley difference, normalised to the lower (right hand side) peak height, is plotted in figure 4(f).The different values scatter around an average of ∼0.25 and a least square line fit is shown to be approximately horizontal.Although optical resolution limits an investigation of the exact geometry around the pointcontact area, the above trends in DF modal diffraction theory simulation and line scans in DF microscopic images provide a signature of the geometry for successfully fabricated devices.
(d).Compared to the valley in figure 4(d), the valley in figure 5(c) is 0.1 × shallower.

Figure 5 (
d) shows normalised light intensity line scans with the Gaussian background removed, extracted from figure 5(b) between the white arrows.Double peaks are observed for lines 1,2, 5 and 6, but are not observed for lines 3, 4. The reduction of valley depth when approaching the centre agrees with the simulation, where a shallow valley is observed around the point-contact area.Figure 5(e) shows an example of the extraction of peaks and valley for line 3 in figure 5(c).A spline fit is applied to the line scan followed by a first order derivative of the line scan.The red crosses in the first derivative plot (bottom figure of figure 5(e)) shows where the ∼0 derivative values are located, corresponding to the x-location of the valley and the right-hand peak respectively.The peak and valley values are then extracted from the normalised power plot (top figure of figure 5(e))